Effect of external tension on the wetting of an elastic sheet

TL;DR

This study investigates how external tension influences the wetting behavior of an elastic sheet, revealing conditions for complete wetting, vesicle formation, and the impact of bending stiffness on the shape and bifurcations of the system.

Contribution

It introduces a comprehensive analysis of wetting on tensioned elastic sheets, including asymptotic descriptions and bifurcation diagrams, highlighting the role of bending stiffness and tension.

Findings

Complete wetting occurs below a critical tension for 0<θ_Y<π/2.

Large tension restores classical partial wetting behavior.

Bending stiffness influences vesicle shape and coexistence with partial wetting.

Abstract

Recent studies of elasto-capillary phenomena have triggered interest in a basic variant of the classical Young-Laplace-Dupr\'e (YLD) problem: The capillary interaction between a liquid drop and a thin solid sheet of low bending stiffness. Here, we consider a two-dimensional model where the sheet is subjected to an external tensile load and the drop is characterized by a well-defined Young's contact angle $\theta_Y$. Using a combination of numerical, variational, and asymptotic techniques, we discuss wetting as a function of the applied tension. We find that, for wettable surfaces with $0<\theta_Y<\pi/2$, complete wetting is possible below a critical applied tension thanks to the deformation of the sheet in contrast with rigid substrates requiring $\theta_Y=0$. Conversely, for very large applied tensions, the sheet becomes flat and the classical YLD situation of partial wetting is recovered. At intermediate tensions, a vesicle forms in the sheet, which encloses most of the fluid and we provide an accurate asymptotic description of this wetting state in the limit of small bending stiffness. We show that bending stiffness, however small, affects the entire shape of the vesicle. Rich bifurcation diagrams involving partial wetting and ``vesicle'' solution are found. For moderately small bending stiffnesses, partial wetting can coexist both with the vesicle solution and complete wetting. Finally, we identify a tension-dependent bendo-capillary length, $\lambda_\text{BC}$, and find that the shape of the drop is determined by the ratio $A/\lambda_\text{BC}^2$, where $A$ is the area of the drop.